Timeline for Are the Yang-Mills equation and its generalization gauge invariant?
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8 events
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| Oct 15, 2014 at 13:07 | comment | added | alxandernashzhang | Ow,great! thanks, best help!I know the relation, but I never use it,I will try to take the calculation about it.If I get the correct answer, I will tell you. | |
| Oct 15, 2014 at 12:57 | comment | added | ACuriousMind♦ | @alxandernashzhang: Yes, the trace is the ordinary matrix trace (though, mathematically, it should be defined through some manipulation of the Killing form, but this need not concern us), and for the adjoint, we usually normalize the generators to fulfill $\mathrm{Tr}(T^a T^b) = \frac{1}{2}\delta^{ab}$ (if I recall correctly), which yields your relation. | |
| Oct 15, 2014 at 12:45 | comment | added | alxandernashzhang | ,you mean,if we take a gauge transformation,the new current $j^{\nu}\prime$ must equal to $Uj^{\nu}U^{-1}$? And I always have another question about the trace in the energy density gauge field: Can I seem the trace as a standard Matrix trace act on the matrix of $F_{\mu\nu}F^{\mu\nu}$, and how to get the relation: $\frac { 1 }{ 2 } { F }_{ \mu \upsilon }^{ a }{ F }_{ a }^{ \mu \upsilon }=tr({ F }_{\mu \upsilon }{ F }^{ \mu \upsilon })$? I teach myself to learn gauge field theory, my supervisor always not investigate the area, so I have some basic problem like this.thanks | |
| Oct 15, 2014 at 12:36 | comment | added | ACuriousMind♦ | @alxandernashzhang: Yes. | |
| Oct 15, 2014 at 12:35 | comment | added | alxandernashzhang | ,you mean,if we take a gauge transformation,the new current $j^{\nu}\prime$ must equal to $Uj^{\nu}U^{-1}$? | |
| Oct 15, 2014 at 12:26 | comment | added | ACuriousMind♦ | @alxandernashzhang: Not really. The equation of motion itself is gauge invariant, since both sides transform in the same representation, so the transformation can be cancelled. Your current should transform as $j_\mu \mapsto Uj_\mu U^{-1}$, if it doesn't, it's the wrong current. Also, please don't post comments as answer - either edit the additional information into the question, or use these comments. | |
| Oct 15, 2014 at 12:01 | comment | added | alxandernashzhang | en, thanks very much! Is your means that this motion equation is not gauge invariant? Please continuously focus this post, I will write down another thing | |
| Oct 15, 2014 at 11:55 | history | answered | ACuriousMind♦ | CC BY-SA 3.0 |